Problem 13 - Elementary Algebra

Question

Perform each multiplication in one step. $$ a^{n} \cdot a^{m} \cdot a^{r} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$a^{n} \cdot a^{m} \cdot a^{r}$$ in one step using the properties of exponents. Answer: $$a^{n+m+r}$$

Step 1: Analyze the problem

Identify the type of problem and the appropriate mathematical technique to apply.

Step 2: Apply the technique and solve

Question: Simplify the expression $$a^{n} \cdot a^{m} \cdot a^{r}$$ in one step using the properties of exponents. Answ.

Step 3: Verify the result

Check the answer by substitution or alternative methods to confirm correctness.

Key Concepts

Exponential Expressions

Understanding how to work with exponential expressions is essential in algebra. An exponential expression is written as a base raised to a power, for instance, in the expression \( a^n \), \( a \) represents the base and \( n \) is the exponent. The exponent tells us how many times to multiply the base by itself. For example, \( a^3 \) means \( a \times a \times a \).

To work with these expressions, it's important to know certain rules. For example, when multiplying two exponents with the same base, you can simply add the exponents. If they have different bases but the same exponent, you can multiply the bases and keep the exponent unchanged. These rules are critical when simplifying complex expressions into a more manageable form.

Multiplying Exponents

When it comes to multiplying exponents, there is a straightforward rule to follow. If you're multiplying exponents with the same base, you keep the base and add the exponents together. The expression \( a^n \times a^m \times a^r \) thus becomes \(a^{n+m+r}\).

The step by step solution presented earlier takes advantage of this rule when it combines \( a^n \times a^m \times a^r \) into \(a^{n+m+r}\) in one step. Remember, this rule only applies when the base remains the same; if you have different bases, the result is not as straightforward and requires a different approach.

Algebraic Simplification

Algebraic simplification is all about making expressions easier to understand and work with. We use the rules of exponentiation to combine terms, reduce fractions, or factor expressions to make them simpler. In our example, instead of multiplying \(a\) by itself multiple times based on the exponents \(n\), \(m\), and \(r\), we use the rule for multiplying exponents to add them up, significantly simplifying the expression.

Good simplification often means finding the shortest expression possible. Once the student grasped the exponentiation rules, algebraic simplification becomes much easier. It is essential, however, to practice these concepts through various exercises to gain confidence and mastery.